Recent developments of Mond-Pečarić method in operator inequalities. Inequalities for bounded selfadjoint operators on a Hilbert space. II
type: Item
instance of: scholarly article
zbMATH Open document ID: 1277.47001
author: Masatoshi Fujii
author: Jadranka Mićić
author: Yuki Seo
author: J. Pečarić
publication date: 2013-07-02T00:00:00Z
Mathematics Subject Classification ID: 47-02
Mathematics Subject Classification ID: 47A63
zbMATH DE Number: 6183114
zbMATH Keywords: Mond-Pečarić method
zbMATH Keywords: operator inequalities
zbMATH Keywords: selfadjoint operators
zbMATH Keywords: operator mean
zbMATH Keywords: Jensen inequality
zbMATH Keywords: Kantorovich inequality
zbMATH Keywords: Araki-Cordes inequality
zbMATH Keywords: Ando-Hiai inequality
zbMATH Keywords: Dunkl-Williams inequality
zbMATH Keywords: arithmetic-geometric mean inequality
zbMATH Keywords: operator monotone function
zbMATH Keywords: operator convexity
MaRDI profile type: Publication
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title: Recent developments of Mond-Pečarić method in operator inequalities. Inequalities for bounded selfadjoint operators on a Hilbert space. II
published in: Monographs in Inequalities
review text: \textit{B. Mond} and \textit{J. E. Pečarić} [Houston J. Math. 20, No. 4, 645--651 (1994; Zbl 0819.47023)] established a method which gives a converse to Jensen's inequality associated with convex functions. Such a method can be used for finding the converse of other inequalities such as the Davis-Choi-Jensen inequality. This interesting book is to be regarded as the second volume of [\textit{T. Furuta} et al., Mond-Pečaric' method in operator inequalities. Inequalities for bounded selfadjoint operators on a Hilbert space. I. Zagreb: ELEMENT (2005; Zbl 1135.47012)], devoted to the Mond-Pečarić method in the field of self-adjoint operators on Hilbert spaces, and includes other results related to operator inequalities. The book consists of eleven chapters ended by some historical notes, an index, and a bibliography consisting of 295 items.NEWLINENEWLINEIn Chapter 1, the authors provide some basic facts about Hilbert space operators. Chapter 2 presents a history of the Kantorovich inequality as well as some ratio and difference type converses of operator versions of Jensen's inequality. There are some overlaps between the two books. One of them is the Furuta inequality and its proof in Chapter 3. This chapter also includes the Heinz inequality. Recently, several reverse Heinz inequalities have appeared in the literature which may be considered in the next volume of this series. In Chapter 4, the authors investigate the Kantorovich type inequalities related to the operator ordering and the chaotic one. Chapter 5 deals with converses of the Araki-Cordes inequality and the Ando-Hiai inequality. The geometric mean of \(n\) operators due to Ando-Li-Mathias and Lawson-Lim is investigated in Chapter 6. In this chapter, a converse of the weighted arithmetic-geometric mean inequality of \(n\) operators is presented. In Chapter 7, the differential-geometrical structure of operators is studied in details. Some properties of Mercer's type operator inequalities are established in Ch…
reviewed by: Mohammad Sal Moslehian
description: scientific article; zbMATH DE number 6183114